Renewal process is a point process where an inter-event time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the renewal process, where the inter-event time distribution alternates between two distributions. We investigate statistical properties of the number of renewals and occupation times for one of the two states in alternating renewal processes. When both means of the inter-event times are finite, the alternating renewal process can reach an equilibrium. On the other hand, an alternating renewal process shows aging when one of the means diverges. We provide analytical calculations for the moments of the number of renewals, occupation time statistics, and the correlation function for several case studies in the inter-event-time distributions. We show anomalous fluctuations for the number of renewals and occupation times when the second moment of inter-event time diverges. When the mean inter-event time diverges, distributional limit theorems for the number of events and occupation times are shown analytically. These are known as the Mittag-Leffler distribution and the generalized arcsine law in probability theory.

翻译：更新过程是一种点过程，其中连续两次更新之间的间隔时间是独立同分布的随机变量。交替更新过程是一种二分过程，是更新过程的轻微推广，其间隔时间分布在两种分布之间交替变化。我们研究了交替更新过程中两个状态之一的更新次数和占据时间的统计特性。当两种间隔时间均值均有限时，交替更新过程可达到平衡态。另一方面，当其中一个均值发散时，交替更新过程表现出老化效应。我们针对几种间隔时间分布案例，解析计算了更新次数的矩、占据时间统计量及相关函数。研究表明，当间隔时间二阶矩发散时，更新次数和占据时间呈现反常波动。当间隔时间均值发散时，我们解析地给出了事件次数和占据时间的分布极限定理，这对应于概率论中的Mittag-Leffler分布和广义反正弦律。